2021 AMC 12B Problems/Problem 19
Problem
Two fair dice, each with at least faces are rolled. On each face of each dice is printed a distinct integer from
to the number of faces on that die, inclusive. The probability of rolling a sum of
is
of the probability of rolling a sum of
and the probability of rolling a sum of
is
. What is the least possible number of faces on the two dice combined?
Solution
Suppose the dice have and
faces, and WLOG
. Since each die has at least
faces, there will always be
ways to sum to
. As a result, there must be
ways to sum to
. There are at most nine distinct ways to get a sum of
, which are possible whenever
. To achieve exactly eight ways,
must have
faces, and
. Let
be the number of ways to obtain a sum of
, then
. Since
,
. In addition to
, we only have to test
, of which both work. Taking the smaller one, our answer becomes
.
Video Solution by OmegaLearn (Using Probability)
~ pi_is_3.14
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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